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Wherever mathematics is being taught or used, Maple has been proven time and again to be the software tool of choice for gaining understanding of advanced concepts or complex technical problems, as well as for providing a single integrated environment in which to develop solutions to those problems.
One area where Maple's power has been greatly appreciated and used extensively is in the modeling of complex dynamic systems, often with a view to developing controllers for these systems.
The Motion Research Group in the Faculty of Engineering at the University of Waterloo, headed by Prof. John McPhee, has been very active in the development of Maple tools for modeling nonlinear, multi-body dynamic systems. These systems cannot generally be analyzed easily and reliably using conventional methods, such as hand derivation coupled with numerical solvers, particularly when dealing with multiple-body, multiple-jointed systems in three dimensions.
"This problem was particularly acute when we were analyzing a Stewart Platform. This is a 6 degree-of-freedom mechanism, used in flight simulators, operated by six hydraulic actuators", said McPhee, "Deriving the equations of motion by hand would have taken us days but, using Maple, we were able to obtain them within hours. And, because the results are symbolic, closed-form expressions for any 6 degree-of-freedom motion for the platform, we could easily perform the inverse dynamic analysis we needed. That is, we could enter the desired motion of the platform and get what forces are required for each actuator. This is crucial for designing the control system for the platform."
After working in the Robotics Group at the Canadian Space Agency, McPhee returned to Waterloo where he and his researchers have been working on two Maple applications, Mego2D (for planar mechanisms) and DynaFlex (for 3D systems). These tools allow engineers to define any generalized multi-body system, using graph theory techniques, from which the equations of motion for the system are automatically generated in exact form.
"Most developers of analysis and simulation software are constrained by the numerical techniques available and end up with huge, generalized solvers, such as ADAMS , DADS, or Working Model," said McPhee. "With Maple and DynaFlex, the computational requirement is very small and you have no restrictions on the nature of the system. Also, by generating the equations of motion for the system you can often gain unique insight into the nature of the dynamics. You can even perform sensitivity analysis on the model by differentiating the equations."
The use of Maple with Mego2D and DynaFlex has been the subject of numerous technical articles (see the Application Center for some of these).
The simple example here shows a quick-return mechanism. The topology (geometry, constraints, masses) of the mechanism can be decomposed into two graphs, one for the translational motion and one for the rotational motion. Using Maple's Graph Theory capabilities, the system can be readily entered into the DynaFlex tool that automatically derives the equations of motion, from which the dynamic behavior of the mechanism can be analyzed.
The use of Maple for modeling dynamic systems has an enormous impact on engineers who need to optimize complex systems or develop control systems. For most engineers, a major part of developing control systems is developing the required understanding of the dynamics of the system to be controlled. Using an analytical approach to developing models in Maple can dramatically reduce the development, freeing the developer to focus on the design of the controller. Not only that, the insight gained when using this approach can the design of the mechanism to be optimized. Maple's modeling approach makes the controller easier to design and results in more robust solution, more quickly and for significantly less cost than with more conventional approaches.
For more information about DynaFlex, Mego2D and how Maple can be used to solve problems in this area, check out our Application Center. |
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